We consider a one-dimensional traffic model with a slow-to-start rule. The initial position of the cars in $$\mathbb {R}$$ R is a Poisson process of parameter $$\lambda $$ λ . Cars have speed 0 or 1 and travel in the same direction. At time zero the speed of all cars is 0; each car waits a mean-one exponential time to switch speed from 0 to 1 and stops when it collides with a stopped car. When the car is no longer blocked, it waits a new exponential time to assume speed one, and so on. We study the saturated regime $$\lambda >1$$ λ > 1 and the critical regime $$\lambda =1$$ λ = 1 , showing that in both regimes all cars collide infinitely often and each car has asymptotic mean velocity $$1/\lambda $$ 1 / λ . In the saturated regime the moving cars form a point process whose intensity tends to 1. The remaining cars condensate in a set of points whose intensity tends to zero as $$1/\sqrt{t}$$ 1 / t . We study the scaling limit of the traffic jam evolution in terms of a collection of coalescing Brownian motions.

中文翻译：

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慢启动流量模型：流量饱和度和扩展限制

我们考虑具有慢启动规则的一维交通模型。$$\mathbb {R}$$ R 中汽车的初始位置是参数 $$\lambda $$ λ 的泊松过程。汽车的速度为 0 或 1，并沿同一方向行驶。在零时刻，所有汽车的速度为 0；每辆汽车都等待平均一指数时间将速度从 0 切换到 1，并在与停止的汽车相撞时停止。当汽车不再被阻塞时，它等待一个新的指数时间以假定速度为 1，依此类推。我们研究了饱和状态 $$\lambda >1$$ λ > 1 和临界状态 $$\lambda =1$$ λ = 1 ，表明在这两种状态下，所有汽车都无限频繁地碰撞，并且每辆车都有渐近平均速度 $ $1/\lambda $$ 1 / λ 。在饱和状态下，移动的汽车形成一个强度趋于 1 的点过程。剩余的汽车凝结在一组强度趋于零的点中，如 $$1/\sqrt{t}$$ 1 / t 。我们根据一系列合并布朗运动研究交通拥堵演变的尺度限制。